GEB class notes Jan 23 Infinite Sets: Cantor, Hilbert, Russell, Gödel The Infinite Hotel Infinite Sums and Zeno’s Paradoxes

Cantor and Counting I Two Sets are “kind of the same” if they have the same number of elements. Infinite Sets are “kind of the same” if they can be put into one-to-one correspondence. The Set of Even Integers is “kind of the same as” the Set of All Integers.

Cantor and Counting II Partitioning the Set of Sets Sets having one element Sets having two elements . . . . . . Infinitely many elements In this context, “kind of the same” is “isomorphic as sets”  FANCY!

Cantor and Counting III Cantor --- How do we distinguish the different kinds of infinite sets? (not a weird question!) “Number of even integers = Number of integers” “Number of powers of 2 = Number of integers” ie., {2, 4, 8,16,32,64, …} is kind of the same as {1, 2, 3, 4, 5, 6, …}

The Infinite Hotel If you’re in Room N, move to Room N+1 always room for one more If you’re in Room N, move to Room 2*N room for infinite busload If you’re in Room N, move to Room 2 N room for infinitely many infinite busloads

Cantor and Counting IV “Number of integer fractions = Number of integers” (1873) “Number of points on line segment is uncountable” (1874) “Number of points in square = Number of points on line segment” (1877) I see it, but I don't believe it!

Cantor Hilbert 1845 - 1918 1862 - 1943

Russell Whitehead 1872 - 1970 1861 - 1947

DOUBTS, FIXES, more DOUBTS Cantor: we don’t know how to count and we can’t handle infinite sets. Russell: what about the set of all sets which are not members of themselves? Russell and Whitehead: Theory of Types and Principia Mathematica Hilbert: desire to demonstrate that Principia Mathematica is both consistent and complete. Godel: powerful and consistent implies incomplete. 1870s 1901 1910 - 1913 1922 - 1927 1931

Theory of Types “The Set of Sets” is an illegal concept. Get around “the Set of Sets” by referring to objects, sets of objects, collections of sets of objects, ensembles of collections of sets of objects, … Hierarchy of language, metalanguage (statements about language), e.g.“German sentences at the end verbs have” metametalanguage (statements about statements about language), …

Let’s catch our breath with a blank slide

An Infinite Sum 1 =  +  1   1

1 =  +  +   1   1

1 =  +  +  +    1   1

1 =  +  +  +  + …   1   1

Follow the Bouncing Ball Infinitely many bounces in finite time http://www.pipey.com/freeflash/gravity.asp

Zeno’s Dichotomy Paradox In going from point A to point B, first you must go halfway. But before reach the halfway point, you must get halfway there. But before you get halfway to the halfway point, you must get halfway there … Therefore motion is impossible!

Response to Zeno IT’S JUST NOT SO, ZENO! If A and B are 1 mile apart and you travel at 1 mile per hour, then (even following Zeno’s description) your total elapsed time is … +  +  +  +  = 1 hour A B     ... IT’S JUST NOT SO, ZENO!

The Tortoise and Achilles How did you handle this using ordinary algebra?

Achilles and the Tortoise recursive formulas for Zeno Achilles and the Tortoise recursive formulas for Zeno A n = A n-1 +  (T n-1 - A n-1) t n = t n-1 + (T n-1 - A n-1 ) ÷ (sA) T n = T n-1 + (t n - t n-1 )*sT The nth stage can be computed from the (n -1)th stage.